Property T C⁎-algebras with amenable tracial states

Abstract

As a non-unital analogue of the main result in [6], we show in this article that if A is a separable quasi-central C⁎-algebra with property T and is nuclear, then there is a sequence of positive integers {nk}k∈N such that the c0-direct sum of the family {Mnk}k∈N of matrix algebras is an ideal JA of A and that A/JA has no tracial state. In particular, a separable C⁎-algebra A is a c0-direct sum of matrix algebras if and only if A is quasi-central, is nuclear, has property T and every unital simple quotient of A has a tracial state. We also show that a separable property TC⁎-algebra A is residually finite dimensional if and only if A admits an amenable faithful tracial state and every ideal of A has non-zero center.